I've just clicked Money > Credit Cards > Find cards and got a list. I also sorted a few with no annual fee. I've not been asked for my address. I would be cautious about revealing my address unless I was actually applying or wanting further details.
If you are not applying and want to get past a postcode box, you can try using someone else's postcode. You should check that your postcode is on the royalmail.com database and, if it's not there (you might be in a new property or somewhere where there have been property changes) you need to contact Royal Mail using the appropriate form on their site as the issue may cause you further problems when you apply and/or with other matters.